Alexander Isak made it many happy Bournemouth returns for Andoni Iraola.
But the Swede was told: “You have to do better.”
Isak, 26, scored his fourth Prem goal of the season to sink the Cherries, crashing home on the half-volley just before the hour.
The Swede ran into the travelling Liverpool fans who gleefully chanted “Isak’s a Red, ole, ole”.
But while Iraola was grateful for the goal, the Liverpool boss urged more from his £125million hitman.
Iraola said: “Alexander took the chance after a quiet afternoon – and that’s what a striker’s job is like.
“He had a couple of situations – a cross he didn’t hit properly, one when his control wasn’t right and then the goal.
“But he has to help us to be closer to the box, to play in the opposition half and to have meaningful possession.
“Then, if the team is playing there, he will receive the rewards in terms of goals. We need that level so that we can play better.”
Isak’s goal was one of just three shots on target for Liverpool, the last of which came from substitute Jason Koumas in stoppage time.
Skipper Virgil van Dijk, whose defence have kept three successive Prem clean sheets, said: “It’s very important for us to get goals, Alexander included.
“It was good win here, a clean sheet and we go to the international break before bracing ourselves for a very intense period.
“Hopefully this team can be very good but let’s all stay calm, keep working and not get ahead of ourselves.
“This was an important win at a difficult ground. We could have done a lot better but we come away with the three points.”
Iraola, who was given a hug by Cherries stars Evanilson and Rayan before kick-off, conceded: “I was disappointed with the first 20-25 minutes. We didn’t have the required intensity.
“Then we started to play better. I know that they never allow you to feel comfortable here. You have to adapt to be able to win this kind of game.
“But we still have to get ready because after the international break the schedule is very difficult.
“We are playing against the top four teams in the Premier League and between those games we have the Champions League and Chelsea in the Carabao Cup, matches every three days. It is going to be very demanding for us.”
Bournemouth are still without a Prem win under new boss Marco Rose, who moaned: “In life, you always get what you deserve.
“You don’t get points for playing well and showing a good attitude.
“We made one error in defence – but that was one error too many. That’s the point.
“It’s not the first time, either. It’s the second, third, fourth, fifth time in the Premier League.
“We need to change our mindset because you need to force wins, in their box and our box.”